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Getting Started

Nemo is a computer algebra package for the Julia programming language, maintained by William Hart, Tommy Hofmann, Claus Fieker, Fredrik Johansson with additional code by Oleksandr Motsak, Marek Kaluba and other contributors.

The features of Nemo so far include:

  • Multiprecision integers and rationals
  • Integers modulo n
  • p-adic numbers
  • Finite fields (prime and non-prime order)
  • Number field arithmetic
  • Algebraic numbers
  • Exact real and complex numbers
  • Arbitrary precision real and complex balls
  • Univariate and multivariate polynomials and matrices over the above

Nemo depends on AbstractAlgebra.jl which provides Nemo with generic routines for:

  • Univariate and multivariate polynomials
  • Absolute and relative power series
  • Laurent series
  • Fraction fields
  • Residue rings
  • Matrices and linear algebra
  • Young Tableaux
  • Permutation groups
  • Characters

Installation

To use Nemo we require Julia 1.6 or higher. Please see https://julialang.org/downloads/ for instructions on how to obtain julia for your system.

At the Julia prompt simply type

julia> using Pkg; Pkg.add("Nemo")

Quick start

Here are some examples of using Nemo.

This example computes recursive univariate polynomials.

julia> using Nemo

julia> R, x = polynomial_ring(ZZ, "x")
(Univariate polynomial ring in x over ZZ, x)

julia> S, y = polynomial_ring(R, "y")
(Univariate polynomial ring in y over univariate polynomial ring, y)

julia> T, z = polynomial_ring(S, "z")
(Univariate polynomial ring in z over univariate polynomial ring, z)

julia> f = x + y + z + 1
z + y + x + 1

julia> p = f^30; # semicolon suppresses output

julia> @time q = p*(p+1);
  0.161733 seconds (79.42 k allocations: 2.409 MiB)

Here is an example using generic recursive ring constructions.

julia> using Nemo

julia> R, x = finite_field(7, 11, "x")
(Finite field of degree 11 and characteristic 7, x)

julia> S, y = polynomial_ring(R, "y")
(Univariate polynomial ring in y over GF(7, 11), y)

julia> T, _ = residue_ring(S, y^3 + 3x*y + 1)
(Residue ring of univariate polynomial ring modulo y^3 + 3*x*y + 1, Map: univariate polynomial ring -> residue ring)

julia> U, z = polynomial_ring(T, "z")
(Univariate polynomial ring in z over residue ring, z)

julia> f = (3y^2 + y + x)*z^2 + ((x + 2)*y^2 + x + 1)*z + 4x*y + 3;

julia> g = (7y^2 - y + 2x + 7)*z^2 + (3y^2 + 4x + 1)*z + (2x + 1)*y + 1;

julia> s = f^12;

julia> t = (s + g)^12;

julia> @time resultant(s, t)
  0.059095 seconds (391.89 k allocations: 54.851 MiB, 5.22% gc time)
(x^10 + 4*x^8 + 6*x^7 + 3*x^6 + 4*x^5 + x^4 + 6*x^3 + 5*x^2 + x)*y^2 + (5*x^10 + x^8 + 4*x^7 + 3*x^5 + 5*x^4 + 3*x^3 + x^2 + x + 6)*y + 2*x^10 + 6*x^9 + 5*x^8 + 5*x^7 + x^6 + 6*x^5 + 5*x^4 + 4*x^3 + x + 3

Here is an example using matrices.

julia> using Nemo

julia> R, x = polynomial_ring(ZZ, "x")
(Univariate polynomial ring in x over ZZ, x)

julia> S = matrix_space(R, 40, 40)
Matrix space of 40 rows and 40 columns
  over univariate polynomial ring in x over ZZ

julia> M = rand(S, 2:2, -20:20);

julia> @time det(M);
  0.080976 seconds (132.28 k allocations: 23.341 MiB, 4.11% gc time)

And here is an example with power series.

julia> using Nemo

julia> R, x = QQ["x"]
(Univariate polynomial ring in x over QQ, x)

julia> S, t = power_series_ring(R, 100, "t")
(Univariate power series ring over univariate polynomial ring, t + O(t^101))

julia> u = t + O(t^100)
t + O(t^100)

julia> @time divexact((u*exp(x*u)), (exp(u)-1));
  0.412813 seconds (667.49 k allocations: 33.966 MiB, 90.26% compilation time)

Building dependencies from source

Nemo depends on the FLINT C library which is installed using binaries by default. With Julia version >= 1.3, the use of this binary can be overridden by putting the following into the file ~/.julia/artifacts/Overrides.toml:

[e134572f-a0d5-539d-bddf-3cad8db41a82]
FLINT = "/prefix/for/libflint"

Experimental threading support for flint

Enabling a threaded version of flint can be done by setting the environment variable NEMO_THREADED=1. To set the actual number of threads, use Nemo.flint_set_num_threads($numberofthreads).